The purpose of these lectures is to present general methods to construct boundary layers both in linear and nonlinear contexts. We will explain how the boundary layer sizes and profiles can be predicted in linear cases, together with some decay estimates. We will illustrate this method with several explicit examples: Ekman layers, reflection of internal waves in a stratified fluid. . . We will also tackle semilinear problems, adding for instance a convection term to the previous examples. Eventually, we will introduce some tools for the study of the Prandtl equation.[-]

The purpose of these lectures is to present general methods to construct boundary layers both in linear and nonlinear contexts. We will explain how the boundary layer sizes and profiles can be predicted in linear cases, together with some decay estimates. We will illustrate this method with several explicit examples: Ekman layers, reflection of internal waves in a stratified fluid. . . We will also tackle semilinear problems, adding for instance a convection term to the previous examples. Eventually, we will introduce some tools for the study of the Prandtl equation.[-]

The purpose of these lectures is to present general methods to construct boundary layers both in linear and nonlinear contexts. We will explain how the boundary layer sizes and profiles can be predicted in linear cases, together with some decay estimates. We will illustrate this method with several explicit examples: Ekman layers, reflection of internal waves in a stratified fluid. . . We will also tackle semilinear problems, adding for instance a convection term to the previous examples. Eventually, we will introduce some tools for the study of the Prandtl equation.[-]

Many hydrodynamic instabilities take place near a solid boundary at high Reynolds number. This reflects into the mathematical theory of the classical Prandtl model for the boundary layer: it exhibits high frequency instabilities, limiting its well-posedness to infinite regularity (Gevrey) spaces. After reviewing shortly this fact, we will turn to the Triple Deck model, a refinement of the Prandtl system that is commonly accepted to be more stable. We will show that this is actually wrong, and that the recent result of analytic well-posedness obtained by Iyer and Vicol is more or less optimal. This is based on joint work with Helge Dietert.[-]